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Nonextensive and superstatistical generalizations of random-matrix theory

  • Topical issue on Generalized Entropies and Non-Linear Kinetics
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Abstract

Random matrix theory (RMT) is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors presented other versions of the theory that keep base invariance on the expense of allowing correlations between matrix elements. This is achieved by starting from non-extensive entropies rather than the standard Shannon entropy, or following the basic prescription of the recently suggested concept of superstatistics. We review these generalizations of RMT and illustrate their value by calculating the nearest-neighbor-spacing distributions and comparing the results of calculation with experiments and numerical-experiments on systems in transition from order to chaos.

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References

  • A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Applied Mathematical Sciences (Springer, New York, 1983)

  • C. Chandre, H.R. Jauslin, Phys. Rep. 365, 1 (2002)

    Google Scholar 

  • J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 67, 617 (1985)

    Google Scholar 

  • S.S.E.H. Elnashaie, S.S. Elshishini, Dynamical Modelling, Bifurcation and Chaotic Behavior of Gas-Solid Catalytic Reactions (Gordon & Breach, Amsterdam, 1996)

  • L.A. Bunimovich, S. Venkatuyiri, Phys. Rep. 290, 81 (1997)

    Google Scholar 

  • M.V. Berry, J. Phys. A 10, 2083 (1977)

  • O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984)

    Google Scholar 

  • M.L. Mehta, Random Matrices, 2nd edn. (Academic, New York, 1991)

  • T. Guhr, A. Müller-Groeling, H.A. Weidenmüller, Phys. Rep. 299, 189 (1998)

  • R. Balian, Nuovo Cim. 57, 183 (1958)

  • N. Rosenzweig, C.E. Porter, Phys. Rev. 120, 1698 (1960)

    Google Scholar 

  • M.S. Hussein, M.P. Sato, Phys. Rev. Lett. 70, 1089 (1993); Phys. Rev. C 47, 2401 (1993)

    Google Scholar 

  • A. Casati, L. Molinari, F. Izrailev, Phys. Rev. Lett. 64, 1851 (1990)

    Google Scholar 

  • Y.V. Fyodorov, A.D. Mirlin, Phys. Rev. Lett. 67, 2405 (1991)

    Google Scholar 

  • A.D. Mirlin, Y.V. Fyodorov, F.M. Dittes, J. Quezada, T.H. Seligman, Phys. Rev. E 54, 3221 (1996)

    Google Scholar 

  • V.E. Kravtsov, K.A. Muttalib, Phys. Rev. Lett. 79, 1913 (1997).

  • F. Evers, A.D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000); Phys. Rev. B 62, 7920 (2000)

    Google Scholar 

  • J. Evans, F. Michael, e-prints arXiv:cond-mat/0207472 and /0208151

  • F. Toscano, R.O. Vallejos, C. Tsallis, Phys. Rev. E 69, 066131 (2004)

  • F.D. Nobre, A.M.C. Souza, Physica A 339, 354 (2004)

    Google Scholar 

  • A.Y. Abul-Magd, Phys. Lett. A 333, 16 (2004)

    Google Scholar 

  • A.C. Bertuola, O Bohigas, M.P. Prato, Phys. Rev. E 70, 065102(R) (2004)

  • A.Y. Abul-Magd, Phys. Rev. E 71, 066207 (2005)

    Google Scholar 

  • A.Y. Abul-Magd, Phys. Lett. A 361, 450 (2007)

    Google Scholar 

  • A.Y. Abul-Magd, Physica A 361, 41 (2006)

    Google Scholar 

  • A.Y. Abul-Magd, Phys. Rev. E 72, 066114 (2005)

    Google Scholar 

  • C. Beck, E.G.D. Cohen, Physica A 322, 267 (2003)

    Google Scholar 

  • E.G.D. Cohen, Physica D 193, 35 (2004)

    Google Scholar 

  • C. Beck, Physica D 193, 195 (2004)

  • C. Beck, Europhys. Lett. 64, 151 (2003)

    Google Scholar 

  • F. Sattin, L. Salasnich, Phys. Rev. E 65, 035106(R) (2003)

  • F. Sattin, Phys. Rev. E 68, 032102 (2003)

    Google Scholar 

  • A. Reynolds, Phys. Rev. Lett. 91, 084503 (2003)

    Google Scholar 

  • M. Ausloos, K. Ivanova, Phys. Rev. E 68, 046122 (2003)

    Google Scholar 

  • C. Beck, Physica A 331, 173 (2004)

  • S. Abe, Phys. Rev. E 66, 046134 (2002)

    Google Scholar 

  • A.M.C. Souza, C. Tsallis, Phys. Lett. A 319, 273 (2003)

    Google Scholar 

  • C. Tsallis, A.M.C. Souza, Phys. Rev. E 67, 026106 (2003); C. Tsallis, A.M.C. Souza, J. Phys. Lett. A 319, 273 (2003)

    Google Scholar 

  • E.T. Jaynes, Phys. Rev. 106, 620 (1957); E.T. Jaynes, Phys. Rev. 108, 171 (1957)

    Google Scholar 

  • C. Tsallis, J. Stat. Phys. 52, 479 (1988)

    Google Scholar 

  • C. Tsallis, Lect. Notes Phys. 560, 3 (2001)

    Google Scholar 

  • Q.A. Wang, Eur. Phys. J. B 26, 357 (2002)

    Google Scholar 

  • C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998)

    Google Scholar 

  • G. Kaniadakis, Physica A 296, 405 (2001)

  • G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)

    Google Scholar 

  • G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)

    Google Scholar 

  • Gradshteyn, I.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1980)

  • C. Beck, E.G.D. Cohen, H.L. Swinney, Phys. Rev. E 72, 026304 (2005)

    Google Scholar 

  • G. Le Caër, R. Delannay, Phys. Rev. E 59, 6281 (1999)

  • K.A. Muttalib, J.R. Klauder, Phys. Rev. E 71, 055101(R) (2005)

  • A.Y. Abul-Magd, B. Dietz, T. Friedrich, A. Richter, Phys. Rev. E 77, 046202 (2008).

    Google Scholar 

  • A. Relaño, J.M.G. Gómez, R.A. Molina, J. Retamosa, E. Faleiro, Phys. Rev. Lett. 89, 244102 (2002); E. Faleiro, J.M.G. Gómez, R.A. Molina, L. Muñoz, A. Relaño, J. Retamosa, Phys. Rev. Lett. 93, 244101 (2004); E. Faleiro, U. Kuhl, R.A. Molina, L. Muñoz, A. Relaño, J. Retamosa, Phys. Lett. A 358, 251 (2006); R.A. Molina, J. Retamosab, L. Muñoz, A. Relaño, E. Faleiro, Phys. Lett. B 644, 25 (2007)

  • J.M.G. Gómez, A. Relaño, J. Retamosa, E. Faleiro, L. Salasnich, M. Vraničar, M. Robnik, Phys. Rev. Lett. 94, 084101 (2005)

  • M.S. Santhanam, J.N. Bandyopadhyay, Phys. Rev. Lett. 95, 114101 (2005)

    Google Scholar 

  • P. Manimaran, P.A. Lakshmi, P.K. Panigrahi, Phys. Rev. E 72, 046120 (2005); J. Phys. A 39, L599 (2006)

    Google Scholar 

  • M.S. Santhanam, J.N. Bandyopadhyay, D. Angom, Phys. Rev. E 73, 015201 (2006)

    Google Scholar 

  • E. Bogomolny, U. Gerald, C. Schmit, Phys. Rev. E 59, R1315 (1999)

  • P.J. Richens, M.V. Berry, Physica (Amsterdam) 2D, 495 (1981)

  • L. Bunimovich, Funct. Anal. Appl. 8, 254 (1974)

    Google Scholar 

  • L.A. Bunimovich, Chaos 11, (2001)

  • M.V. Berry, M. Robnik, J. Phys. A 17, 2413 (1984)

    Google Scholar 

  • T. Brody, Lett. Nuovo Cim. 7, 482 (1973)

    Google Scholar 

  • B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, F. Schäfer, Phys. Rev. E 75, 035203(R) (2007)

  • H.P. Baltes, E.R. Hilf, Spectra of Finite Systems (Bibliographisches Institut Mannheim, 1975)

  • Y. Gu, K.W. Yu, Z.R. Yang, Phys. Rev. E 65, 046129 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering Sciences, Sinai University, 068, El Arish, Egypt

    A. Y. Abul-Magd

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Abul-Magd, A. Nonextensive and superstatistical generalizations of random-matrix theory. Eur. Phys. J. B 70, 39–48 (2009). https://doi.org/10.1140/epjb/e2009-00153-0

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